积分恒等式与广义$m$-拟爱因斯坦流形的刚性

中文摘要 AI 辅助

我们建立了闭广义$m$-拟爱因斯坦流形$(M^n,g,X,\lambda)$的微分与积分恒等式。这些恒等式给出了共形性、Killing条件和平凡性的判据，并为若干刚性现象提供了统一框架。此后，我们将积分Bochner公式重新表述为Witten型Hodge能量恒等式。其在$m=-2$处四次项的消去，在自然符号条件下导致了广义情形下的平凡性定理，并特别证明了当$\lambda\leq0$为常数时，$m=-2$情形下最近Colling--Dunajski猜想。漂移拉普拉斯算子的一个恒等式给出了已知$m\leq-n$平凡性结果的新证明，并将其推广到广义$m$-拟爱因斯坦流形。在常数$\lambda$情形下，我们进一步推导了扭曲算子的微分恒等式，该恒等式特别突出了$m=-4$的值。结合谱与最大值原理论证，这导致了$\lambda\leq0$时的平凡性，从而证明了$m=-4$情形下的Colling--Dunajski猜想。最后，作为我们刻画共形位势场何时为Killing场的判据的补充，我们在附录中构造了一个闭广义$m$-拟爱因斯坦流形，其位势场是共形的但非Killing的。

英文摘要

We prove that every closed $m$-quasi-Einstein manifold $(M^n,g,X,λ)$ with constant $λ\leq0$ is trivial whenever $m\leq-2$. This establishes the Colling--Dunajski conjecture in the range $m\leq-2$ and, for $n\geq5$, extends the previously known range $m\leq2-n$. This result is placed within a broader study of closed generalized $m$-quasi-Einstein manifolds $(M^n,g,X,λ)$. We establish differential and integral identities for such manifolds. These identities yield criteria for conformality, the Killing condition, and triviality, and provide a unified framework for several rigidity phenomena. After this, we recast the integrated Bochner formula as a Witten-type Hodge-energy identity. The cancellation of its quartic term at $m=-2$ leads to a triviality theorem in the generalized setting under a natural sign condition on the integral of $\langle X,\nablaλ\rangle$. An identity for the drift Laplacian gives a new proof of the previously known triviality result for $m\leq-n$ and extends it to generalized $m$-quasi-Einstein manifolds under a pointwise sign condition on $\langle X,\nablaλ\rangle$. Finally, complementing our criterion characterizing when a conformal potential field is Killing, we exhibit in the appendix a closed generalized $m$-quasi-Einstein manifold whose potential field is conformal but non-Killing.

发表机构

• Universidade Federal de Pernambuco（伯南布哥联邦大学）
• Universidade Federal de Alagoas（阿拉戈阿斯联邦大学）

机构由 AI 辅助整理，请以论文原文为准。